Communications in Contemporary Mathematics

Let X be a non-compact complete manifold (or a graph) which admits a quasi-pole and has bounded local geometry. Suppose that X is Gromov-hyperbolic and the diameters (for a fixed Gromov metric) of the connected components of X(∞) have a positive lower bound. Under these assumptions we show that X has positive Cheeger isoperimetric constant. Examples are also constructed to show that the Cheeger constant h(X) may be zero if any of the above assumption on X is removed.

Applications of this isoperimetric estimate include the solvability of the Dirichlet problem at infinity for non-compact Gromov-hyperbolic manifolds X above. In addition, we show that the Martin boundary ∂_ΔX of such a space X is homeomorphic to the geometric boundary X(∞) of X at infinity.

A maximum/minimum principle for weighted W^1,2 solutions to the Tricomi problem with L² right hand side and homogeneous boundary data is established for normal Tricomi domains. In addition, the existence and uniqueness of such generalized solutions is established for arbitrary L² right hand sides in normal domains which satisfy a convexity condition near the parabolic boundary points.

In this paper we develop a method for studying tight contact structures on lens spaces. We then derive uniqueness and non-existence statements for tight contact structures with certain (half) Euler classes on lens spaces.

We prove that the dimension of the space of primitive Vassiliev invariants of degree n grows — as n tends to infinity — faster than for any . This solves the so-called Kontsevich–Bar–Natan conjecture.

The proof relies on the use of the weight systems coming from the Lie algebra (N). In fact, we show that our bound is — up to a multiplication by a rational function in n — the best possible that one can get with (N)-weight systems.